The surjectivity conjecture for Springer-fiber restriction maps

Let $\fg$ be a classical Lie algebra of type $B$, $C$, or $D$, with dual Lie algebra $\fg^\vee$, and let $e^\vee\in\fg^\vee$ be nilpotent with partition $\lambda^\vee$. Let $\fl^\vee$ be a Levi subalgebra containing a regular representative associated with $e^\vee$, and let

$$i^*:H^*(\cB^\vee)\longrightarrow H^*(\spr)$$

be the pullback from the flag variety of $\fg^\vee$ to the relevant Springer fiber. The surjectivity conjecture. The map $i^*$ is surjective exactly in the following cases: (1) $\fg^\vee=\fsp_{2n}$ and $e^\vee$ is regular in $\lv=\fsp_{2a}\times\prod_{i=1}^{k}\fgl_{b_i}$ with $b_i\in\{2a+1,2a,2a-1,1\}$; (2) $\fg^\vee=\fso_{2n+1}$ and $e^\vee$ is regular in $\lv=\fso_{2a+1}\times\prod_{i=1}^{k}\fgl_{b_i}$ with $b_i\in\{2a+2,2a+1,2a\}$; or (3) $\fg^\vee=\fso_{2n}$ and $\lambda^\vee$ has one of the five displayed forms in the source statement. This conjecture is attributed to HMK2024 and is used to formulate the paper's conditional Hikita results; the paper proves only certain cases.